Push mail notifications not working?

Hello, I'm new to iOS, coming from Android. My iPhone 5s has the mail app connected to my gmail account. In the settings, I have push notifications on but I'm not getting notifications/sound for them when I receive them. I have to manually check the app every time. However, if I send a mail to myself on the phone I do receive the notification and sound.
Apple TV

Fetch will do just that any time you open the Mail app... or got to settings>> mail...>> fetch new data>> scroll down and set 15 minutes, or 30, or hourly... or manual if you want that... and make sure your gmail account is set to fetch on that screen as well... or select manual if you want that.